MATRIX AND ITS APPLICATIONS
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The introduction and development of the notion of a matrix and the
subject of linear algebra followed the development of determinants,
which arose from the study of coefficients of systems of linear
equations. Leibnitz, one of the founder of calculus, used determinant in
1963 and Cramer presented his determinant based formula for solving
systems of linear equation (today known as Cramer’s rule) in 1750.
The first implicit use of matrices occurred in Lagrange’s work on
bilinear form in late 1700. Lagrange desired to characterize the maxima
and minima of multi-variant functions. His method is now known as the
method of Lagrange multipliers. In order to do this he first required
the first order partial derivation to be 0 and additionally required
that a condition on the matrix of second order partial derivatives
holds; this condition is today called positive or negative
definiteness, although Lagrange did not use matrices explicitly.
Gauss developed elimination around 1800 and used it to solve least
square problem in celestial computations and later in computations to
measure the earth and it’s surface (the branch of applied mathematics
concerned with measuring or determining the shape of the earth or with
locating exactly points on the earth’s surface is called Geodesy). Even
though Gauss name is associated with this technique eliminating
variable from system of linear equations there were earlier work on
Chinese manuscripts from several centuries earlier have been
found that explains how to solve a system of three equations in three
unknown by “Guassian” elimination. For years Gaussian elimination was
considered part of the development of geodgesy, not mathematics. The
first appearance of Gaussian-Jordan elimination in print was in a
handbook on geodesy written by Wihelm Jordan. Many people incorrectly
assume that the famous mathematician, Camille Jordan is the Jordan in
For matrix algebra to fruitfully develop one needed both proper
notation and proper definition of matrix multiplication. Both needs
were met at about the same time in the same place. In 1848 in England,
J.J Sylvester first introduced the term “matrix”, which was the Latin
word for “womb” as a name for an array of numbers.
Matrix algebra was nurtured by the work of Arthur Cayley in 1855.
Cayley studied multiplication so that the matrix of coefficient for the
composite transformation ST is the product of the matrix S times the
matrix T. He went on to study the algebra of these composition
including matrix inverses. The famous Cayley-Hamilton theorem which
asserts that a square matrix is a root of it’s characteristics’
polynomial was given by Cayley in his 1858 memoir on the theory of
matrices. The use of single letter “A to represent a matrix was crucial
to the development of matrix algebra. Early in the development, the
formular det(AB) = det (A) det(B) provided a connection between matrix
algebra and determinants. Cayley wrote “There would be many things to
say about this theory of matrices which should, it seems to me, precede
the theory of determinants”.
Mathematicians also attempted to develop for algebra of vectors
but there was no natural definition of the product of two vectors that
held in arbitrary dimensions. The first vector algebra that involved a
non- commutative vector product (that is V x W need not equal W x V)
was proposed by Hermann Grassman in his book – Ausedehnungslehre
(1844). Grossmann’s text also introduced the product of a column matrix
and a row matrix, which resulted in what is now called a simple or a
rank one matrix. In the late 19th century the American mathematical
physicist, Willard Gibbs published his famous treatise on vector
analysis. In that treatise Gibbs represented general matrices, which he
called dyadics as sum of simple matrices, which Gibbs called dyads.
Later the physicist, P.A.M. Dirac introduced the term “bracket” for
what we now call the “scalar product” of a “bar” (row) vector times a
“ket” (column) vector and the term “ket-bra” for the product of a ket
times a bra, resulting in what we now call a simple matrix, as above.
Our convention of identifying column matrices and vector was introduced
by physicists in the 20th century.
Matrices continued to be closely associated with linear
transformations. By 1900, they were just a finite dimensional sub case
of the emerging theory of linear transformations. The modern definition
of a vector space was introduced by Peano in 1888. Abstract vector
space whose elements were function soon followed. There was renewed
interests in matrices, particularly on the numerical analysis of
matrices, John Von Neumann and Herman Goldstein introduced condition
numbers in analyzing round – off errors in 1947. Alan Turing and Von
Neumann, the 20th century giants in the development of stored – program
computers. Turning introduced the LU decomposition of a matrix
in 1948. The L is a lower triangular matrix with I’s on the diagonal
and the U is an echelon matrix. It is common to use LU decomposition in
the solution of n sequence of systems of linear equations,
each having the same co-efficient matrix. The benefit of the QR
decomposition was realized a decade later. The Q is a matrix whose
column are orthogonal vector and R is a square upper triangular
invertible matrix with positive entities on its diagonal.
The QR factorization is used in computer algorithms for various
computations, such as solving equations and find eigenvalues. Frobenius
in 1878 wrote an important work on matrices on linear substitutions
and bilinear forms, although he seemed unaware of Cayley’s work.
However be proved important results in canonical matrices as
representatives of equivalences classes of matrices. He cites Kronecker
and Weiserstrases having considered special cases of his results in
1868 and 1874 respectively.
Frobenius also proved the general result that a matrix satisfies
it’s characteristic equation. This 1878 paper by Frobenius also
contains the definition of the rank of a matrix, which he used in his
work on canonical forms and the definition of orthogonal matrices.
An axiomatic definition of a determinant was used by Weierstrass in
his lectures and after his death, it was published in 1903 in the note
on determinant theory. In the same year Kronecker’s lectures on
determinants were also published after his death. With these two
publications, the modern theory of determinants was in place but
matrix theory took slightly longer to become a fully accepted theory.
An important early text which brought matrices into their proper place
within mathematics was introduction to higher algebra by Bocher in
1907. Turnbull and Aitken wrote influential text in the 1930s and
Missky’s; “An introduction to linear algebra” in 1955 saw matrix theory
to reach its present major role as one of the most important
undergraduate mathematics topic.
MATRIX AND ITS APPLICATIONS
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